Drop an ordinary ice cube into a glass of water and it floats. Now freeze some heavy water, the liquid that India's pressurised heavy water reactors use by the hundreds of tonnes. An ice cube of heavy water sinks to the bottom of the glass. Heavy water looks, tastes and boils almost like ordinary water. Its molecules are still two hydrogen atoms and one oxygen atom.
The two kinds of nucleon
The chapter on the Bohr atom found that an atom has a tiny nucleus that holds all its positive charge and nearly all its mass. The nucleus is itself made of two kinds of particle. The proton carries a charge \(+e\). The neutron carries no charge at all. Protons and neutrons are together called nucleons.
Nucleons are far too light to weigh in kilograms comfortably. So nuclear physics uses the atomic mass unit, one twelfth of the mass of a carbon-12 atom: \(1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg}\). The proton has mass \(1.007276\,\text{u}\) and the neutron \(1.008665\,\text{u}\), a little more. The electron has only \(0.000549\,\text{u}\), about 1836 times less than the proton.
Two numbers name a nucleus
Two whole numbers describe a nucleus. The atomic number \(Z\) is the number of protons. It fixes the charge \(Ze\) of the nucleus and the number of electrons in the neutral atom. So it fixes the chemistry: every nucleus with \(Z = 26\) is iron. The mass number \(A\) is the total number of nucleons. The neutron number follows at once: \(N = A - Z\).
A nuclide is one kind of nucleus, fixed by its atomic number \(Z\) and its mass number \(A\). It is written \({}^{A}_{Z}\text{X}\), where X is the chemical symbol of the element. In words it is written X-\(A\), as in uranium-235 or U-235.
Read a few symbols. Sodium, \({}^{23}_{11}\text{Na}\), has 11 protons and \(23 - 11 = 12\) neutrons. Iron-56, \({}^{56}_{26}\text{Fe}\), has 26 protons and 30 neutrons. Uranium-235, \({}^{235}_{92}\text{U}\), has 92 protons and 143 neutrons. The lower number is often left out, because the symbol already says it: U always means \(Z = 92\).
The mass number \(A\) is a count, not a mass. Uranium-235 has 235 nucleons, but its atom has a mass of \(235.04\,\text{u}\), not exactly 235 u. Later in the chapter, the small differences between a count and a mass turn out to hold all of nuclear energy.
How many neutrons are there in a nucleus of uranium-238, \({}^{238}_{92}\text{U}\)?
In 1920 Rutherford guessed that a nucleus must hold some neutral particle as heavy as a proton. Nobody could find one, because a neutral particle leaves no track. In 1932 James Chadwick studied a strange radiation that came out of beryllium struck by alpha particles. It knocked protons out of paraffin wax with far more energy than any gamma ray could give them. Chadwick showed that it was a stream of neutral particles about as heavy as protons: neutrons. He received the Nobel Prize in 1935.
Families of nuclei
Three numbers describe a nucleus, \(Z\), \(N\) and \(A\), and two nuclei may share any one of them. Each kind of sharing has a name. Fig 1.1 shows one family of each kind, with every proton in red and every neutron in grey.
Isotopes have the same \(Z\) and different \(N\). They are the same element, with the same electrons and the same chemistry, but different masses. Hydrogen has three: ordinary hydrogen \({}^{1}_{1}\text{H}\), deuterium \({}^{2}_{1}\text{H}\) and tritium \({}^{3}_{1}\text{H}\), which is radioactive. Chlorine is a mixture of chlorine-35 and chlorine-37, which is why its average atomic mass is 35.5 u.
Isobars have the same \(A\) and different \(Z\). Carbon-14, with 6 protons and 8 neutrons, and nitrogen-14, with 7 of each, are isobars. They are different elements that happen to hold 14 nucleons each. Isotones have the same \(N\): carbon-13, nitrogen-14 and oxygen-15 all have 7 neutrons.
Now the ice cube can be explained. Heavy water is \(\text{D}_2\text{O}\): each hydrogen atom in it is deuterium, with a neutron beside its proton. Each molecule therefore carries two extra neutrons, about 20.0 u against 18.0 u for ordinary water. Heavy water is 11 % denser than ordinary water, and its ice is denser than ordinary water too, so the cube sinks.
Carbon-14 (\(Z = 6\)) and nitrogen-14 (\(Z = 7\)) are best described as
Which of these statements are true?
How big is a nucleus?
Rutherford's alpha particles gave only an upper limit. A 7.7 MeV alpha particle turned back \(29.5\,\text{fm}\) from a gold nucleus without touching it, as the chapter on the Bohr atom showed. Here \(1\,\text{fm}\), one femtometre, is \(10^{-15}\,\text{m}\). Faster probes were needed to reach the surface. Electrons of several hundred MeV scatter from the charge inside a nucleus, and their pattern maps where that charge sits.
The measurements give a simple rule. The volume of a nucleus is proportional to its mass number. Twice the nucleons fill twice the volume, like marbles poured into a bag. Since a sphere's volume goes as \(R^3\), the radius grows only as the cube root of \(A\).
Try a few values of \(A\) in the figure above. Hydrogen, \(A = 1\), has \(R = 1.2\,\text{fm}\). Aluminium-27 has \(A^{1/3} = 3\), so \(R = 3.6\,\text{fm}\). Uranium-238 has 238 nucleons, 238 times as many, yet its radius is only \(7.4\,\text{fm}\), about six times larger. A 238-fold volume needs only a sixfold radius.
Find the radius of a nucleus of iron-56. Then find the ratio of the radii of two nuclei with \(A = 216\) and \(A = 27\).
\(R = R_0A^{1/3}\) with \(R_0 = 1.2\,\text{fm}\).
\(R_{56} = 1.2 \times 56^{1/3}\), and \(R_{216}/R_{27} = (216/27)^{1/3}\).
\(56^{1/3} = 3.83\), so \(R_{56} = 4.6\,\text{fm}\). The ratio is \(8^{1/3} = 2\).
Iron-56 has a radius of \(4.6\,\text{fm}\), and the nucleus with 216 nucleons is twice as wide as the one with 27. Values used: \(R_0 = 1.2\,\text{fm}\), \(1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg}\).
A graph of \(R\) against \(A^{1/3}\) is therefore a straight line through the origin. Its slope is \(R_0\). Measured radii of many nuclei fall close to such a line, which is how \(R_0 = 1.2\,\text{fm}\) was found.
A nucleus has a radius of \(6.0\,\text{fm}\). Take \(R_0 = 1.2\,\text{fm}\). What is its mass number?
Nucleus P has 8 times the mass number of nucleus Q. The ratio of their radii \(R_P/R_Q\) is
One density for every nucleus
The second readout in the figure above did not move. Every nucleus, from hydrogen to uranium, has the same density. The reason is in the two rules just met. The mass of a nucleus is about \(A\) times the mass of one nucleon, and its volume is \(A\) times the volume per nucleon. The factor \(A\) cancels.
\[ \rho = \dfrac{Am}{\tfrac{4}{3}\pi R^3} \]
Why. the mass is \(A\) nucleons of mass \(m\) each; the nucleus is a sphere of radius \(R\).
\[ \rho = \dfrac{Am}{\tfrac{4}{3}\pi R_0^3 A} \]
Why. put in \(R^3 = R_0^3A\), the cube of \(R = R_0A^{1/3}\).
\[ \rho = \dfrac{3m}{4\pi R_0^3} \]
Read it. \(A\) cancels. The density depends only on the nucleon's mass and \(R_0\).
With \(m = 1.66 \times 10^{-27}\,\text{kg}\) and \(R_0 = 1.2 \times 10^{-15}\,\text{m}\), the nuclear density comes out as \(2.3 \times 10^{17}\,\text{kg/m}^3\). Water has a density of \(10^3\,\text{kg/m}^3\). Nuclear matter is \(2.3 \times 10^{14}\) times denser. A teaspoon of it, \(5\,\text{cm}^3\), would have a mass of more than a billion tonnes.
Ordinary matter is so much lighter because atoms are nearly empty. All the mass of an atom sits in its nucleus, but the atom is tens of thousands of times wider. In a neutron star, the collapsed core of a large star, matter really is packed at nuclear density. A whole star of this kind is only about 20 km across.
How does the density of a uranium-238 nucleus compare with the density of a helium-4 nucleus?
The Sun has a mass of \(2 \times 10^{30}\,\text{kg}\). If it were squeezed to nuclear density, \(2.3 \times 10^{17}\,\text{kg/m}^3\), what would its radius be? Answer in kilometres.
The Earth has a mass of \(6 \times 10^{24}\,\text{kg}\). If it were squeezed to nuclear density, \(2.3 \times 10^{17}\,\text{kg/m}^3\), estimate its radius in metres.
- Reading the mass number as the mass of the nucleus in u.\(A\) counts nucleons. The actual mass differs slightly from \(A\) u, and that difference matters later.
- Calling C-14 and N-14 isotopes because both have 14 nucleons.Isotopes share \(Z\). Nuclei sharing \(A\) are isobars.
- Taking the radius as proportional to \(A\).The volume is proportional to \(A\), so \(R \propto A^{1/3}\): eight times the nucleons, twice the radius.
- Thinking a heavy nucleus is denser than a light one.Mass and volume both grow as \(A\). Every nucleus has the same density, \(2.3 \times 10^{17}\,\text{kg/m}^3\).